7.4.3.

Are the means equal?

The procedure known as the Analysis of Variance or
ANOVA is used to test hypotheses concerning means when we
have several populations.

The Analysis of Variance (ANOVA)

The ANOVA procedure is one of the most powerful statistical
techniques

ANOVA is a general technique that can be used to test the hypothesis
that the means among two or more groups are equal, under the
assumption that the sampled populations are normally distributed.

A couple of questions come immediately to mind: what means?
and why analyze variances in order to derive conclusions
about the means?

Both questions will be answered as we delve further into the subject.

Introduction to ANOVA

To begin, let us study the effect of temperature on a passive
component such as a resistor. We select three different
temperatures and observe their effect on the resistors. This
experiment can be conducted by measuring all the participating
resistors before placing \(n\)
resistors each in three different
ovens.

Each oven is heated to a selected temperature. Then we measure the
resistors again after, say, 24 hours and analyze the responses,
which are the differences between before and after being subjected
to the temperatures. The temperature is called a factor.
The different temperature settings are called levels. In
this example there are three levels or settings of the factor
Temperature.

What is a factor?

A factor is an independent treatment variable whose settings
(values) are controlled and varied by the experimenter. The
intensity setting of a factor is the level.

Levels may be quantitative numbers or, in many cases,
simply "present" or "not present" ("0" or "1").

The one-way ANOVA

In the experiment above, there is only one factor, temperature,
and the analysis of variance that we will be using to analyze the
effect of temperature is called a one-way or one-factor
ANOVA.

The two-way or three-way ANOVA

We could have opted to also study the effect of positions in the
oven. In this case there would be two factors, temperature and
oven position. Here we speak of a two-way or two-factor
ANOVA. Furthermore, we may be interested in a third factor, the
effect of time. Now we deal with a three-way or
three-factor ANOVA. In each of these ANOVA techniques we test a
variety of hypotheses of equality of means (or average responses
when the factors are varied).

Hypotheses that can be tested in an ANOVA

First consider the one-way ANOVA. The null hypothesis is: there
is no difference in the population means of the different levels of
factor \(A\) (the only factor).

The alternative hypothesis is: the means are not the same.

For the two-way ANOVA, the possible null hypotheses are:

There is no difference in the means of factor \(A\)

There is no difference in means of factor \(B\)

There is no interaction between factors \(A\) and \(B\)

The alternative hypothesis for cases 1 and 2 is: the means are not
equal.

The alternative hypothesis for case 3 is: there is an interaction
between \(A\) and \(B\).

For the three-way ANOVA, the main effects are factors \(A\), \(B\), and \(C\), and
the two-factor interactions are \(AB\), \(AC\), and \(BC\). There is also a
three-factor interaction, \(ABC\).

For each of the seven cases the null hypothesis is the same:
there is no difference in means, and the alternative hypothesis is
the means are not equal.

The \(n\)-way ANOVA

In general, the number of main effects and interactions can be
found by the following expression:
$$ N = \left( \begin{array}{c} n \\ 0 \end{array} \right) +
\left( \begin{array}{c} n \\ 1 \end{array} \right) +
\left( \begin{array}{c} n \\ 2 \end{array} \right) + \ldots +
\left( \begin{array}{c} n \\ n \end{array} \right) \, . $$
The first term is for the overall mean, and is always 1. The second
term is for the number of main effects. The third term is for the
number of two-factor interactions, and so on. The last term is for
the \(n\)-factor
interaction and is always 1.